We study the spectrum of the normalized Laplace operator of a connected graph $Gamma$. As is well known, the smallest nontrivial eigenvalue measures how difficult it is to decompose $Gamma$ into two large pieces, whereas the largest eigenvalue controls how close $Gamma$ is to being bipartite. The smallest eigenvalue can be controlled by the Cheeger constant, and we establish a dual construction that controls the largest eigenvalue. Moreover, we find that the neighborhood graphs $Gamma[l]$ of order $lgeq2$ encode important spectral information about $Gamma$ itself which we systematically explore. In particular, the neighborhood graph method leads to new estimates for the smallest nontrivial eigenvalue that can improve the Cheeger inequality, as well as an explicit estimate for the largest eigenvalue from above and below. As applications of such spectral estimates, we provide a criterion for the synchronizability of coupled map lattices, and an estimate for the convergence rate of random walks on graphs.
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